no, for example Trenton and Albany are not the largest cities in NJ and NY
No
Yes it is always the largest city in that state
No
No
No it does not have to be the largest city because of that
No.Seattle is largest in Washington and the capital is Olympia
No, not always. New York City is the largest city in New York State (by population it's the largest city in the United States) but Albany is the state capital. Los Angeles is the largest city in California (and second-largest by population in the U.S.) but Sacramento is the state capital. Detroit is the largest city in Michigan, but Lansing is the state capital. Chicago is the largest city in Illinois, but Springfield is the state capital.
No, it is not.
No
No, it is not. Consider Florida, Texas, California or Nevada.
No, there are many states with larger cities than their capital.
No, the state's capital is not always the largest city in the state. While some state capitals are also the largest city, there are many cases where a smaller city serves as the capital. For example, in New York, Albany is the capital even though it is not the largest city (that would be New York City).